When Jack was one year old his mother bought a packet of $24$ candles for his birthday cake.

That year she put $1$ candle on Jack's cake. When he was two he had $2$ candles and when he was three he had $3$ candles, and so on.

One day Jack's little sister Kate was born. She had $1$ candle on her first birthday cake, $2$ candles on her second birthday cake, and so on.

The candles were finished on one of Jack's birthdays with just enough left.

How old was Jack when Kate was born? And how old was each of them when the candles finally ran out?